Properties

Label 8T31
Order \(64\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $31$
CHM label :  $[2^{4}]E(4)$
Parity:  $-1$
Primitive:  No
Generators:   (4,8), (1,3)(2,8)(4,6)(5,7), (1,8)(2,3)(4,5)(6,7)
$|\Aut(F/K)|$:  $2$
Low degree resolvents:  
2: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1
4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2
8: 4T3, 4T3, 4T3, 4T3, 4T3, 4T3, 8T3
16: 8T9, 8T9, 8T9
32: 8T18

Subfields

Degree 2: $C_2$, $C_2$, $C_2$

Degree 4: $V_4$

Low degree siblings

8T29a, 8T29b, 8T29c, 8T29d, 8T29e, 8T29f, 8T31b, 16T127, 16T128a, 16T128b, 16T128c, 16T129a, 16T129b, 16T129c, 16T147, 16T149a, 16T149b, 16T149c, 16T149d, 16T149e, 16T149f, 16T150a, 16T150b, 16T150c
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $(4,8)$
$ 2, 2, 1, 1, 1, 1 $ $2$ $2$ $(3,7)(4,8)$
$ 2, 2, 1, 1, 1, 1 $ $2$ $2$ $(2,6)(4,8)$
$ 2, 2, 1, 1, 1, 1 $ $2$ $2$ $(2,6)(3,7)$
$ 2, 2, 2, 1, 1 $ $4$ $2$ $(2,6)(3,7)(4,8)$
$ 2, 2, 2, 2 $ $4$ $2$ $(1,2)(3,4)(5,6)(7,8)$
$ 4, 2, 2 $ $8$ $4$ $(1,2)(3,4,7,8)(5,6)$
$ 4, 4 $ $4$ $4$ $(1,2,5,6)(3,4,7,8)$
$ 2, 2, 2, 2 $ $4$ $2$ $(1,3)(2,4)(5,7)(6,8)$
$ 4, 2, 2 $ $8$ $4$ $(1,3)(2,4,6,8)(5,7)$
$ 4, 4 $ $4$ $4$ $(1,3,5,7)(2,4,6,8)$
$ 2, 2, 2, 2 $ $4$ $2$ $(1,4)(2,3)(5,8)(6,7)$
$ 4, 2, 2 $ $8$ $4$ $(1,4,5,8)(2,3)(6,7)$
$ 4, 4 $ $4$ $4$ $(1,4,5,8)(2,3,6,7)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$

Group invariants

Order:  $64=2^{6}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [64, 138]
Character table:  
      2  6  4  5  5  5  4  4  3  4  4  3  4  4  3  4  6

        1a 2a 2b 2c 2d 2e 2f 4a 4b 2g 4c 4d 2h 4e 4f 2i
     2P 1a 1a 1a 1a 1a 1a 1a 2b 2i 1a 2c 2i 1a 2d 2i 1a
     3P 1a 2a 2b 2c 2d 2e 2f 4a 4b 2g 4c 4d 2h 4e 4f 2i

X.1      1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
X.2      1 -1  1  1  1 -1 -1  1 -1 -1  1 -1  1 -1  1  1
X.3      1 -1  1  1  1 -1 -1  1 -1  1 -1  1 -1  1 -1  1
X.4      1 -1  1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1
X.5      1 -1  1  1  1 -1  1 -1  1  1 -1  1  1 -1  1  1
X.6      1  1  1  1  1  1 -1 -1 -1 -1 -1 -1  1  1  1  1
X.7      1  1  1  1  1  1 -1 -1 -1  1  1  1 -1 -1 -1  1
X.8      1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1  1
X.9      2  .  2 -2 -2  . -2  .  2  .  .  .  .  .  .  2
X.10     2  .  2 -2 -2  .  2  . -2  .  .  .  .  .  .  2
X.11     2  . -2 -2  2  .  .  .  .  .  .  . -2  .  2  2
X.12     2  . -2 -2  2  .  .  .  .  .  .  .  2  . -2  2
X.13     2  . -2  2 -2  .  .  .  . -2  .  2  .  .  .  2
X.14     2  . -2  2 -2  .  .  .  .  2  . -2  .  .  .  2
X.15     4 -2  .  .  .  2  .  .  .  .  .  .  .  .  . -4
X.16     4  2  .  .  . -2  .  .  .  .  .  .  .  .  . -4